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Archard equation

The Archard equation is a simple model used to describe sliding wear and is based around the theory of asperity contact.

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Equation

$Q = \frac {KW}H$

where:

Q is the total volume of wear debris produced per unit distance moved

W is the total normal load

H is the hardness

K is a dimensionless constant

Derivation

The equation can be derived by first examining the behavior of a single asperity.

The local load $\, \delta W$ , supported by an asperity, assumed to have a circular cross-section with a radius $\, a$ , is:

$\delta W = P \pi {a^2} \,\!$

where P is the yield pressure for the asperity, assumed to be deforming plastically. P will be close to the indentation hardness, H, of the asperity.

If the volume of wear debris, $\, \delta V$ , for a particular asperity is a hemisphere sheared off from the asperity, it follows that:

$\delta V = \frac 2 3 \pi a^3$

This fragment is formed by the material having slid a distance 2a

Hence, $\, \delta Q$ , the wear volume of material produced from this asperity per unit distance moved is:

$\delta Q = \frac {\delta V} {2a} = \frac {\pi a^2} 3 \equiv \frac {\delta W} {3P} \approx \frac {\delta W} {3H}$ making the approximation that $\,P \approx H$

However, not all asperities will have had material removed when sliding distance 2a. Therefore, the total wear debris produced per unit distance moved, $\, Q$ will be lower than the ratio of W to 3H. This is accounted for by the addition of a dimensionless constant K, which also incorporates the factor 3 above. These operations produce the Archard equation as given above.